Problem: Let $x$ and $y$ be nonzero real numbers. Let $m$ and $M$ be the minimium value and maximum value of
\[\frac{|x + y|}{|x| + |y|},\]respectively.  Find $M - m.$
Solution: Both $|x + y|$ and $|x| + |y|$ are nonnegative, so $\frac{|x + y|}{|x| + |y|}$ must be nonnegative.  When $x = 1$ and $y = -1,$
\[\frac{|x + y|}{|x| + |y|} = \frac{0}{2} = 0,\]so this is clearly the minimum.

On the other hand, by the Triangle Inequality, $|x| + |y| \ge |x + y|,$ so
\[\frac{|x + y|}{|x| + |y|} \le 1.\]Equality occurs when $x = y,$ so the maximum is 1.

Therefore, $M - m = 1 - 0 = \boxed{1}.$